(last updated: 12:59:09 PM, November 08, 2020)
Definition 4.3.1 A set \(E \subset\mathbb R\) is said to be closed if and only if every accumulation point of \(E\) is in \(E\).
Definition 4.3.2 A set \(E \subset\mathbb R\) is said to be open if and only if for each \(x \in E\) there exists a neighborhood \(I\) of \(x\) such that \(I\) is entirely contained in \(E\).
Theorem 4.3.3 A set\(E \subset\mathbb R\) is closed if and only if \(\mathbb R \setminus E\) is open.
Theorem 4.3.4 If a function \(f\) is continuous on a closed and bounded interval \([a, b]\), then \(f\) is bounded on \([a, b]\).
Theorem 4.3.5 (Extreme Value Theorem) If a function \(f\) is continuous on a closed and bounded interval \([a, b]\), then \(f\) attains its maximum and minimum values on \([a , b]\).
Theorem 4.3.6 (Bolzano’s Intermediate Value Theorem) If a function \(f\) is continuous on \([a, b]\) and if \(k\) is a real number between \(f(a)\) and \(f(b)\), then there exists a real number \(c \in (a, b)\) such that \(f (c) = k\).
Definition 4.3.7 A function \(f: D\subset\mathbb R \rightarrow\mathbb R\) satisfies the intermediate value property on \(D\) if and only if for every \(x_1, x_2 \in D\) with \(x_1 < x_2\) and any real constant \(k\) between \(f (x_1)\) and \(f (x_2)\) there exists at least one constant \(c \in (x_1, x_2)\) such that \(f (c) = k\).
Theorem 4.3.11 If a function \(f: D \rightarrow\mathbb R\) is a continuous injection and \(D = [a, b]\), then \(f^{-1}: f(D) \rightarrow D\) is continuous.
\[ \diamond \S \diamond \]